Open Problems Problem 1. Determine the consistency strength of the statement u 2 = ω 2, where u 2 is the second uniform indiscernible. Best known bounds: Con(there is a strong cardinal) Con(u 2 = ω 2 ) Con(there is a Woodin cardinal with a measurable above it) Remark: The difficulty is that we don t know how to use the hypothesis u 2 = ω 2 to build larger models, even if we were given a measurable to make sense of the core model K. Related Problem: Determine the consistency strength of the statement every real has a sharp + every subset of ω L(R) 1 in L(R) is constructible from a real. Problem 2. Assume PD. C 3 is the largest countable Π 1 3-set of reals. Is it true that C 3 = {x M 2 R x is 1 3 definable from a mastercode of M 2 }? C 1 = {x x is 1 1 equivalent to a mastercode of L} C 2 = R L The reals in C 3 are Turing cofinal in C 4 = M 2 R. Problem 3. Working in ZFC, how large can Θ L(R) be? ω V 1 < Θ L(R) Con(ω V 2 < Θ L(R) ) if e.g. u 2 = ω 2 But what about ω V 3 < Θ L(R)? Variant: Assume there are arbitrarily large Woodin cardinals. Is it possible that there is a universally Baire well-ordering of ordertype ω V 3? Problem 4. Assume AD + and assume that there is no iteration strategy for a countable mouse with a superstrong. Let a R OD. Does there exist a countable iterable mouse M such that a M? Remark: If there is an iteration strategy Σ for a countable mouse with a superstrong then we work in an initial segment of the Wadge hierarchy < W Σ. (Woodin) If we replace the hypothesis with no iteration strategy for a countable mouse satisfying the AD R hypothesis then the conclusion follows. This is the best result known. 1
Problem 5. Suppose M 1 (x) exists for all sets x. K will be closed under the operation x M 1 (x) and any model closed under this operation will be Σ1 3 correct. Is K Σ 1 4 correct under the hypotheses that M 1 (x) exists for all x and that there is no inner model with 2 Woodins? Remark: (Steel) Assume that K exists below a Woodin cardinal (e.g. ORD is measurable) and assume that there is a measurable and that there is no inner model with 1 Woodin. Then K is Σ 1 3 correct. Conjectured improvement to Steel s Theorem: Assume that x(x exists) and that there is no inner model with a Woodin. Is K Σ 1 3 correct? There are partial results in this direction (Woodin and others). Problem 6. Assume that x(m 1 (x) exists) and that there is a least Π1 3 singleton, z, that is not in the least inner model, N, closed under the operation x M 1 (x). Does N exist and is z 1 3 isomorphic to N? Remark: This would give Σ 1 4 correctness for K in the case that K doesn t go beyond N. The second clause in the above conclusion is an instance of problem 2. Problem 7. a) Let M be a countable, transitive structure that is elementarily embeddable into some V α. Is M (ω 1 + 1) iterable? b) (An instance of CBH) For every countable iteration tree on V of limit length such that every extender used is countably closed from the model from which it was taken, is there a cofinal well-founded branch? Note: Countably closed means that ω Ult(V, E) Ult(V, E). In any L[ E] model UBH is true. (Woodin) If you drop countable closure then CBH (i.e. full CBH) is false. Problem 8. Let L[ E] be an extender model such that every countable structure elementarily embeddable into a level of the model is (ω 1 + 1) iterable (so that many forms of condensation hold). Characterize (in terms of large cardinal axioms) all successor cardinals (κ + ) L[ E] of L[ E] such that L[ E] = (every stationary subset S κ + Cof(ω) reflects). Variant: Characterize all (κ + ) L[ E] such that L[ E] = (every stationary subset S κ + Cof(< κ) reflects). Problem 9. What is the consistency strength of λ for some singular λ? Best upper bound known: κ(κ is κ +ω strongly compact) Contrast this with the best upper bound known for ℵ ω : there is a measurable subcompact. 2
Problem 10. Let j : V M be elementary, assume ORD is measurable and assume there is no proper class inner model with a Woodin. Is j K an iteration map? (Schindler) This is true if ω M M. Problem 11. a) Rate the consistency strength of the following statement: Let I be a simply definable σ-ideal.then the statement Every Σ 1 2 (projective) I-positive set has a Borel I-positive subset holds in every generic extension. b) Assume 0 #. Is it possible to force over V a real x such that R V [G] L[x] is I-positive? Remark: Here definable means simple, i.e., countable sets, Lebegue null sets, meager sets, and etc. Problem 12. Prove that if V = W [r] for some real r, V and W have the same cofinalities, W = CH, and V = 2 ℵ0 = ℵ 2, then there is an inner model with ℵ 2 many measurables. (Shelah) Under the above hypothesis, there is an inner model with a measurable. Problem 13. Investigate the following ZFC-model: HOD V [G] where G is generic over V for Coll(ω, < ORD). In particular, Does CH hold in this model? Problem 14. Assume 0-Pistol doesn t exist. Suppose κ is Mahlo and κ (Sing) fails. a) Must κ be a measurable in K? b) Suppose, in addition, that GCH holds below κ. Is there an inner model with a strong cardinal? c) Can GCH hold? (Woodin) CON(o(κ) = κ ++ + ɛ) CON( κ is Mahlo and κ fails ). (Zeman) If κ is Mahlo and κ (Sing) fails then for all λ < κ there is δ < κ such that K = o(δ) > λ. 3
Problem 15. a) Assume there is no proper class inner model with a Woodin cardinal. Must there exist a set-iterable extender model which satisfies weak covering? b) Does CON( ZFC + NS ω1 is ℵ 2 -saturated ) imply CON(ZFC + there is a Woodin cardinal )? Problem 16. Let M be the minimal fully iterable extender model which satisfies there is a Woodin cardinal κ which is a limit of Woodin cardinals. Let D be the derived model of M bellow κ. Does D = θ is regular? Problem 17. Determine the consistency strength of incompatible models of AD + (i.e. there are A and B such that L(A, R) and L(B, R) satisfy AD + but L(A, B, R) = AD). (Neeman and Woodin) Upper Bound: Woodin limit of Woodin cardinals. (Woodin) Lower Bound: AD R + DC. Problem 18. Is HOD L(R) θ a normal iterate of M ω δ 0 where δ 0 is the least Woodin of M ω? If not, is there a normal iterate Q of HOD L(R) fixing θ such that Q θ is a normal iterate of every countable iterate of M ω δ 0? (Neeman) The answer to the first question is almost no. Problem 19. Assume V = L(R) + AD. Let Γ be a Π 1 1-like scaled pointclass (i.e., closed under R and non-self-dual). Let δ = sup{ < :< is a pwo in = Γ Γ }. Then, is Γ closed under unions of length < δ? (Kechris-Martin) Known for Π 1 3. (Jackson) Known for Π 1 2n+5. 4
Problem 20. Is there an inner model M of L[0 # ] such that 0 # M, 0 # M[G], and (M[G], G) = ZFC, where G is P-generic over M for some M-definable class-forcing? 5